Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $a = \dfrac{-5p + 30}{p + 8} \times \dfrac{p^2 + 10p + 16}{p^2 - 6p} $
Solution: First factor the quadratic. $a = \dfrac{-5p + 30}{p + 8} \times \dfrac{(p + 8)(p + 2)}{p^2 - 6p} $ Then factor out any other terms. $a = \dfrac{-5(p - 6)}{p + 8} \times \dfrac{(p + 8)(p + 2)}{p(p - 6)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ -5(p - 6) \times (p + 8)(p + 2) } { (p + 8) \times p(p - 6) } $ $a = \dfrac{ -5(p - 6)(p + 8)(p + 2)}{ p(p + 8)(p - 6)} $ Notice that $(p - 6)$ and $(p + 8)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ -5\cancel{(p - 6)}(p + 8)(p + 2)}{ p\cancel{(p + 8)}(p - 6)} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $a = \dfrac{ -5\cancel{(p - 6)}\cancel{(p + 8)}(p + 2)}{ p\cancel{(p + 8)}\cancel{(p - 6)}} $ We are dividing by $p - 6$ , so $p - 6 \neq 0$ Therefore, $p \neq 6$ $a = \dfrac{-5(p + 2)}{p} ; \space p \neq -8 ; \space p \neq 6 $